A332793 - OEIS

%I #22 Apr 04 2023 12:34:54

%S 1,2,-3,8,-5,-6,-7,32,0,-10,-11,-24,-13,-14,15,128,-17,0,-19,-40,21,

%T -22,-23,-96,0,-26,0,-56,-29,30,-31,512,33,-34,35,0,-37,-38,39,-160,

%U -41,42,-43,-88,0,-46,-47,-384,0,0,51,-104,-53,0,55,-224,57,-58,-59,120

%N a(1) = 1; a(n) = n * Sum_{d|n, d < n} (-1)^(n/d) * a(d) / d.

%H Amiram Eldar, <a href="/A332793/b332793.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} (-1)^k * k * A(x^k).

%F Dirichlet g.f.: 1 / (zeta(s-1) * (1 - 2^(2 - s))).

%F a(n) = Sum_{d|n} A327268(d).

%F Multiplicative with a(2^e) = 2^(2*e-1), and a(p^e) = -p if e=1 and 0 for e>1, for odd primes p. - _Amiram Eldar_, Dec 02 2020

%t a[1] = 1; a[n_] := n Sum[If[d < n, (-1)^(n/d) a[d]/d, 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 60}]

%t terms = 60; A[_] = 0; Do[A[x_] = x + Sum[(-1)^k k A[x^k], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] // Rest

%t f[p_, e_] := If[p == 2, p^(2*e - 1), -p*Boole[e == 1]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 02 2020 *)

%Y Cf. A002129, A038838 (positions of 0's), A055615, A067856, A327268, A361987.

%Y Partial sums give A361982.

%Y Dirichlet inverse of A181983.

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 24 2020